 1.1.1: Show that in any base b 2, the sum of any three singledigit number...
 1.1.2: Show that any binary integer is at most four times as long as the c...
 1.1.3: A dary tree is a rooted tree in which each node has at most d chil...
 1.1.4: Show thatlog(n!) = (n log n).(Hint: To show an upper bound, compare...
 1.1.5: Unlike a decreasing geometric series, the sum of the harmonic serie...
 1.1.6: Prove that the gradeschool multiplication algorithm (page 24), whe...
 1.1.7: How long does the recursive multiplication algorithm (page 25) take...
 1.1.8: Justify the correctness of the recursive division algorithm given i...
 1.1.9: Starting from the definition of x y mod N (namely, that N divides x...
 1.1.10: Show that if a b (mod N) and if M divides N then a b (mod M).
 1.1.11: Is 41536 94824 divisible by 35?
 1.1.12: What is 222006 (mod 3)?
 1.1.13: Is the difference of 530,000 and 6123,456 a multiple of 31?
 1.1.14: Suppose you want to compute the nth Fibonacci number Fn, modulo an ...
 1.1.15: Determine necessary and sufficient conditions on x and c so that th...
 1.1.16: The algorithm for computing ab mod c by repeated squaring does not ...
 1.1.17: Consider the problem of computing xyfor given integers x and y: we ...
 1.1.18: Compute gcd(210, 588) two different ways: by finding the factorizat...
 1.1.19: The Fibonacci numbers F0, F1, . . . are given by the recurrence Fn+...
 1.1.20: Find the inverse of: 20 mod 79, 3 mod 62, 21 mod 91, 5 mod 23.
 1.1.21: How many integers modulo 113 have inverses? (Note: 113 = 1331.)
 1.1.22: Prove or disprove: If a has an inverse modulo b, then b has an inve...
 1.1.23: Show that if a has a multiplicative inverse modulo N, then this inv...
 1.1.24: If p is prime, how many elements of {0, 1, . . . , pn 1} have an in...
 1.1.25: Calculate 2125 mod 127 using any method you choose. (Hint: 127 is p...
 1.1.26: What is the least significant decimal digit of 171717 ? (Hint: For ...
 1.1.27: Consider an RSA key set with p = 17, q = 23, N = 391, and e = 3 (as...
 1.1.28: In an RSA cryptosystem, p = 7 and q = 11 (as in Figure 1.9). Find a...
 1.1.29: Let [m] denote the set {0, 1, . . . , m 1}. For each of the followi...
 1.1.30: The gradeschool algorithm for multiplying two nbit binary numbers...
 1.1.31: Consider the problem of computing N! = 1 2 3 N.(a) If N is an nbit...
 1.1.32: A positive integer N is a power if it is of the form qk, where q, k...
 1.1.33: Give an efficient algorithm to compute the least common multiple of...
 1.1.34: On page 38, we claimed that since about a 1/n fraction of nbit num...
 1.1.35: Wilsons theorem says that a number N is prime if and only if(N 1)! ...
 1.1.36: Square roots. In this problem, well see that it is easy to compute ...
 1.1.37: The Chinese remainder theorem.(a) Make a table with three columns. ...
 1.1.38: To see if a number, say 562437487, is divisible by 3, you just add ...
 1.1.39: Give a polynomialtime algorithm for computing abcmod p, given a, b...
 1.1.40: Show that if x is a nontrivial square root of 1 modulo N, that is, ...
 1.1.41: Quadratic residues. Fix a positive integer N. We say that a is a qu...
 1.1.42: Suppose that instead of using a composite N = pq in the RSA cryptos...
 1.1.43: In the RSA cryptosystem, Alices public key (N, e) is available to e...
 1.1.44: Alice and her three friends are all users of the RSA cryptosystem. ...
 1.1.45: RSA and digital signatures. Recall that in the RSA publickey crypt...
 1.1.46: Digital signatures, continued. Consider the signature scheme of Exe...
Solutions for Chapter 1: Algorithms with numbers
Full solutions for Algorithms  1st Edition
ISBN: 9780073523408
Solutions for Chapter 1: Algorithms with numbers
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Since 46 problems in chapter 1: Algorithms with numbers have been answered, more than 17079 students have viewed full stepbystep solutions from this chapter. Chapter 1: Algorithms with numbers includes 46 full stepbystep solutions. Algorithms was written by and is associated to the ISBN: 9780073523408. This textbook survival guide was created for the textbook: Algorithms , edition: 1.

Analytic study
A study in which a sample from a population is used to make inference to a future population. Stability needs to be assumed. See Enumerative study

Average
See Arithmetic mean.

Bias
An effect that systematically distorts a statistical result or estimate, preventing it from representing the true quantity of interest.

Contingency table.
A tabular arrangement expressing the assignment of members of a data set according to two or more categories or classiication criteria

Continuity correction.
A correction factor used to improve the approximation to binomial probabilities from a normal distribution.

Control chart
A graphical display used to monitor a process. It usually consists of a horizontal center line corresponding to the incontrol value of the parameter that is being monitored and lower and upper control limits. The control limits are determined by statistical criteria and are not arbitrary, nor are they related to speciication limits. If sample points fall within the control limits, the process is said to be incontrol, or free from assignable causes. Points beyond the control limits indicate an outofcontrol process; that is, assignable causes are likely present. This signals the need to ind and remove the assignable causes.

Convolution
A method to derive the probability density function of the sum of two independent random variables from an integral (or sum) of probability density (or mass) functions.

Correlation coeficient
A dimensionless measure of the linear association between two variables, usually lying in the interval from ?1 to +1, with zero indicating the absence of correlation (but not necessarily the independence of the two variables).

Counting techniques
Formulas used to determine the number of elements in sample spaces and events.

Covariance matrix
A square matrix that contains the variances and covariances among a set of random variables, say, X1 , X X 2 k , , … . The main diagonal elements of the matrix are the variances of the random variables and the offdiagonal elements are the covariances between Xi and Xj . Also called the variancecovariance matrix. When the random variables are standardized to have unit variances, the covariance matrix becomes the correlation matrix.

Cumulative normal distribution function
The cumulative distribution of the standard normal distribution, often denoted as ?( ) x and tabulated in Appendix Table II.

Decision interval
A parameter in a tabular CUSUM algorithm that is determined from a tradeoff between false alarms and the detection of assignable causes.

Empirical model
A model to relate a response to one or more regressors or factors that is developed from data obtained from the system.

Enumerative study
A study in which a sample from a population is used to make inference to the population. See Analytic study

Estimator (or point estimator)
A procedure for producing an estimate of a parameter of interest. An estimator is usually a function of only sample data values, and when these data values are available, it results in an estimate of the parameter of interest.

Experiment
A series of tests in which changes are made to the system under study

Exponential random variable
A series of tests in which changes are made to the system under study

Extra sum of squares method
A method used in regression analysis to conduct a hypothesis test for the additional contribution of one or more variables to a model.

Factorial experiment
A type of experimental design in which every level of one factor is tested in combination with every level of another factor. In general, in a factorial experiment, all possible combinations of factor levels are tested.

Geometric mean.
The geometric mean of a set of n positive data values is the nth root of the product of the data values; that is, g x i n i n = ( ) = / w 1 1 .